Demand for a certain Commodity

[confusedbycalc]

The demand for a certain commodity is

D(x)  =  1000e−.03x

units per month when the market price is x dollars per unit.

(a) At what rate is the consumer expenditure E(x) = xD(x) changing with respect to price x when the price is equal to $160 dollars?
(b) At what price does consumer expenditure stop increasing and begin to decrease?
(c) At what price does the rate of consumer expenditure begin to increase?

I am not sure but I got -31.27 for a) but i really have no idea how to go about this question.

 

[HallsofIvy]

The demand for a certain commodity is

D(x)  =  1000e−.03x

What is "e" here? (I suspect it is not 2.718...

units per month when the market price is x dollars per unit.

(a) At what rate is the consumer expenditure E(x) = xD(x) changing with respect to price x when the price is equal to $160 dollars?

With D(x)= 1000e- 0.3x, E(x)= 1000ex- 0.3x^2. The rate of change of that is the derivative E'(x)= 1000e- 0.6x.

(b) At what price does consumer expenditure stop increasing and begin to decrease?

As long as E' is positive the consumer expenditure is increasing. It is decreasing when E' is negative. To change from positive to negative, E' has to become 0.

(c) At what price does the rate of consumer expenditure begin to increase?

when is E' positive?

I am not sure but I got -31.27 for a) but i really have no idea how to go about this question.